Bordered Heegaard Floer Homology.

Cover -- Title page -- Chapter 1. Introduction -- 1.1. Background -- 1.2. The bordered Floer homology package -- 1.3. On gradings -- 1.4. The case of three-manifolds with torus boundary -- 1.5. Previous work -- 1.6. Further developments -- 1.7. Organization -- Acknowledgments -- Chapter 2. \textalt{\Ainf}A-infty structures -- 2.1. \textalt{\Ainf}A-infty algebras and modules -- 2.2. \textalt{\Ainf}A-infty tensor products -- 2.3. Type \textalt{ }D structures -- 2.4. Another model for the \textalt{\Ainf}A-infty tensor product -- 2.5. Gradings by non-commutative groups -- Chapter 3. The algebra associated to a pointed matched circle -- 3.1. The strands algebra \textalt{\Alg( , )}A(n,k) -- 3.2. Matched circles and their algebras -- 3.3. Gradings -- Chapter 4. Bordered Heegaard diagrams -- 4.1. Bordered Heegaard diagrams: definition, existence, and uniqueness -- 4.2. Examples of bordered Heegaard diagrams -- 4.3. Generators, homology classes and \textalt{\spin^{ }}spin-c structures -- 4.4. Admissibility criteria -- 4.5. Closed diagrams -- Chapter 5. Moduli spaces -- 5.1. Overview of the moduli spaces -- 5.2. Holomorphic curves in \textalt{Σ×[0,1]×\RR}Sigma × [0,1] × R -- 5.3. Holomorphic curves in \textalt{\RR× ×[0,1]×\RR} R × Z × [0,1] × R -- 5.4. Compactifications via holomorphic combs -- 5.5. Gluing results for holomorphic combs -- 5.6. Degenerations of holomorphic curves -- 5.7. More on expected dimensions -- Chapter 6. Type \textalt{ }D modules -- 6.1. Definition of the type \textalt{ }D module -- 6.2. \textalt{\bdy²=0}Boundary-squared is zero -- 6.3. Invariance -- 6.4. Twisted coefficients -- Chapter 7. Type \textalt{ }A modules -- 7.1. Definition of the type \textalt{ }A module -- 7.2. Compatibility with algebra -- 7.3. Invariance -- 7.4. Twisted coefficients -- Chapter 8. Pairing theorem via nice diagrams.

Chapter 9. Pairing theorem via time dilation -- 9.1. Moduli of matched pairs -- 9.2. Dilating time -- 9.3. Dilating to infinity -- 9.4. Completion of the proof of the pairing theorem -- 9.5. A twisted pairing theorem -- 9.6. An example -- Chapter 10. Gradings -- 10.1. Algebra review -- 10.2. Domains -- 10.3. Type \textalt{ }A structures -- 10.4. Type \textalt{ }D structures -- 10.5. Refined gradings -- 10.6. Tensor product -- Chapter 11. Bordered manifolds with torus boundary -- 11.1. Torus algebra -- 11.2. Surgery exact triangle -- 11.3. Preliminaries on knot Floer homology -- 11.4. From \textalt{\CFDa}CFDˆ to \textalt{\HFKm}HFK- -- 11.5. From \textalt{\CFKm}CFK- to \textalt{\CFDa}CFDˆ: Statement of results -- 11.6. Generalized coefficient maps and boundary degenerations -- 11.7. From \textalt{\CFKm}CFK- to \textalt{\CFDa}CFDˆ: Basis-free version -- 11.8. Proof of Theorem 11.26 -- 11.9. Satellites revisited -- Appendix A. Bimodules and change of framing -- A.1. Statement of results -- A.2. Sketch of the construction -- A.3. Computations for \textalt{3}3-manifolds with torus boundary -- A.4. From \textalt{\HFK}HFK to \textalt{\CFDa}CFDˆ for arbitrary integral framings -- Bibliography -- Index of Definitions -- Back Cover.

The authors construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra and the other of which (type A) is an \mathcal A_\infty module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the \mathcal A_\infty tensor product of the type D module of one piece and the type A module from the other piece is \widehat{HF} of the glued manifold. As a special case of the construction, the authors specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for \widehat{HF}. The authors relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.